Math Fundamentals


Spline curves

  • The term spline curve now refers to any composite curve formed with polynomial sections satisfying any specified continuity conditions at the boundary of the pieces.

  • A spline surface can be described with two sets of spline curves.

  • Splines are used to design curve and surface shapes, to digitize drawings, and to specify animation paths for the objects or the camera position in a scene.

Interpolation and approximation splines

  • We specify a spline curve by giving a set of coordinate positions, called control points

  • In Figure 1, the resulting curve is said to interpolate the set of control points.

  • In Figure 2, the resulting curve is said to approximate the set of control points

Bézier Spline Curves

  • This spline approximation method was developed by the French engineer Pierre Bézier for use in the design of Renault automobile bodies.

  • Bézier splines have a number of properties that make them highly useful and convenient for curve and surface design.

  • A Bézier curve section can be fitted to any number of control points, although some graphic packages limit the number of control points to four.

  • The degree of the Bézier polynomial is determined by the number of control points to be approximated and their relative position.

Challenge: Dynamic Bézier Curve-Generating

OpenGL complete sample code / Processing

  • Design a program that will dynamically draw a bézier curve as the control points are being defined with mouse clicks
  • First and second clicks will define the first and final control points
  • Third and fourth clicks will define 2 dynamic control points
  • Dynamic control points can be moved with the mouse (click and drag)

More Spline Curve Algorithms

  • B-Spline Curves
  • Beta-Spline Curves
  • Rational Splines

Check chapter 13 of the book

What's next




Resources and Credits

This material is genereated thanks to some extracts from following resources:

  • Computer Graphics with OpenGL (Chapter ) by Donald D. Hearn/M. Pauline Baker, Warren Carithers, 4th Edition


  • Obed N Muñoz Reynoso
    • Cloud Software Engineer
    • obed.n.munoz@gmail | tec |